IR-MO 2003 Problem 11

assume that X is a set of n number.and $0\leq k\leq n$ .the maximum number of permutation which acting on $X$ st every two of them have at least k component in common,is $a_{n,k}$ .and the maximum nuber of permutation st every two of them have at most k component in common,is $b_{n,k}$ . a)proeve that : $a_{n,k}\cdot b_{n,k-1}\leq n!$ b)assume that p is prime number,determine the exact value of $a_{p,2}$ .